Determine how many solutions exist for the system of equations. ${3x+y = 9}$ ${6x+2y = 18}$
Solution: Convert both equations to slope-intercept form: ${3x+y = 9}$ $3x{-3x} + y = 9{-3x}$ $y = 9-3x$ ${y = -3x+9}$ ${6x+2y = 18}$ $6x{-6x} + 2y = 18{-6x}$ $2y = 18-6x$ $y = 9-3x$ ${y = -3x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+9}$ ${y = -3x+9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${3x+y = 9}$ is also a solution of ${6x+2y = 18}$, there are infinitely many solutions.